They beat Manchester United in the FA Cup but that adventure came to an end with a quarter-final defeat by Tottenham.
West Ham announced Rednapp had left the manager's hotseat "by mutual consent" - but it has emerged there was a dispute over the club's potential summer transfer fund.
Redknapp and Brown disagreed over what the club needed to be competitive next season. Redknapp felt this left him with no option.
Redknapp had hoped to strengthen his squad with the remainder of the £18m he raked in from the sale of Rio Ferdinand to Leeds United last season, but he said they needed more.
But it is understood there was a belief inside the club that Redknapp had not spent wisely.
Redknapp said: "I said we need to spend £12-14 million on three or four quality players just to avoid standing still."
Charlton Athletic boss Alan Curbishley is the firm favourite to take over - although West Ham will face fierce opposition inside The Valley if they attempt to land him.
Redknapp's departure came after a season which saw the London club struggle in the league.
West Ham only secured their top-flight status with victory against Southampton in the penultimate game of the campaign.
They beat Manchester United in the FA Cup but that adventure came to an end with a quarter-final defeat by Tottenham.
Redknapp, a former player at Upton Park, had been in charge since August 1994 and was the second longest-serving manager in the top-flight behind Sir Alex Ferguson.
Redknapp's assistant Frank Lampard senior also left the club and Glenn Roeder will take over as caretaker manager for West Ham's final game of the season against Middlesbrough.
But former Hammers striker Tony Cottee was surprised by Redknapp's departure.
He said: "They've gone from being a team that fights relegation every season to, until this season, an established top 10 Premiership club."
Ferdinand, who was sold by West Ham to Leeds for £18m, said: "Harry gave me my chance in football and I will always appreciate that.
Redknapp had said that the Upton Park side needed three more players to help secure a finish in the top-half of the Premiership next season.
Doubts surround Redknapp's future plans, but he has already been linked as the next Southampton boss because of his south-coast connection.
Rednapp revealed: "A lot of people have linked me to the Southampton vacancy but I haven't spoken to anyone at The Dell."